Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

EPSRC Postdoctoral Research Fellow







614Huxley BuildingSouth Kensington Campus





Publication Type

10 results found

Feyzbakhsh S, Li C, 2021, Higher rank Clifford indices of curves on a K3 surface, SELECTA MATHEMATICA-NEW SERIES, Vol: 27, ISSN: 1022-1824

Journal article

Feyzbakhsh S, 2020, Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing, Journal für die reine und angewandte Mathematik, Vol: 2020, Pages: 101-137, ISSN: 0075-4102

Let C be a curve of genus g=11 or g≥13 on a K3 surface whose Picard group is generated by the curve class [C]. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai’s program to this situation: we show how to reconstruct the K3 surface containing the curve C as a Fourier–Mukai transform of a Brill–Noether locus of vector bundles on C.

Journal article

Feyzbakhsh S, Thomas RP, 2019, An application of wall-crossing to Noether-Lefschetz loci, Publisher: arXiv

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Giesekerconjecture of Bayer-Macr\`{i}-Toda (such as $\mathbb P^3$, the quinticthreefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If$c_1(L)$ is a primitive cohomology class then we show it has very negativesquare.

Working paper

Bayer A, Beentjes S, Feyzbakhsh S, Hein G, Martinelli D, Rezaee F, Schmidt Bet al., The desingularization of the theta divisor of a cubic threefold as a moduli space

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaveson a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$is smooth and of dimension four. Moreover, the Abel-Jacobi map to theintermediate Jacobian of $X$ maps it birationally onto the theta divisor$\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to thesingular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of theTorelli theorem for cubic threefolds, which says that $X$ can be recovered fromits Kuznetsov component $\operatorname{Ku}(X) \subset\mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of thedescription of the singularity of the theta divisor, and thus of the classicalTorelli theorem for cubic threefolds, i.e., that $X$ can be recovered from itsintermediate Jacobian.

Journal article

Feyzbakhsh S, Thomas RP, Rank $r$ DT theory from rank $0$

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalisedDT invariants counting Gieseker semistable sheaves of any rank $r\ge1$ on $X$in terms of those counting sheaves of rank 0 and pure dimension 2. The basic technique is to reduce the ranks of sheaves by replacing them bythe cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing tohandle their stability.

Journal article

Feyzbakhsh S, Thomas RP, Rank $r$ DT theory from rank $1$

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistablesheaves of any rank $r$ on $X$ in terms of those counting sheaves of rank 1. Bythe MNOP conjecture they are therefore determined by the Gromov-Witteninvariants of $X$.

Journal article

Feyzbakhsh S, Thomas RP, Curve counting and S-duality

We work on a projective threefold $X$ which satisfies the Bogomolov-Giesekerconjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quinticthreefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ aresmooth bundles over Hilbert schemes of ideal sheaves of curves and points in$X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressingcurve counts (and so ultimately Gromov-Witten invariants) in terms of counts ofD4-D2-D0 branes. These latter invariants are predicted to have modularproperties which we discuss from the point of view of S-duality andNoether-Lefschetz theory.

Journal article

Feyzbakhsh S, Pertusi L, Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

We prove a general criterion which ensures that a fractional Calabi--Yaucategory of dimension $\leq 2$ admits a unique Serre-invariant stabilitycondition, up to the action of the universal cover of$\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component$\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all theknown stability conditions on $\text{Ku}(X)$ are invariant with respect to theaction of the Serre functor and thus lie in the same orbit with respect to theaction of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As anapplication, we show that the moduli space of stable Ulrich bundles of rank$\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macr\`iand Stellari.

Journal article

Feyzbakhsh S, An effective restriction theorem via wall-crossing and Mercat's conjecture

We prove an effective restriction theorem for stable vector bundles $E$ on asmooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors$D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than$2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgelandstability conditions which we also use to reprove Camere's result on slopestability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.

Journal article

Feyzbakhsh S, Mukai's Program (reconstructing a K3 surface from a curve) via wall-crossing, II

Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$.Mukai's program seeks to recover $X$ from $C$ by exhibiting it as aFourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We usewall-crossing in the space of Bridgeland stability conditions to prove this forgenus $\ge14$. This paper deals with the case $g-1$ prime left over from PaperI.

Journal article

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